Skip to main content
Contents Index
Dark Mode Prev Up Next
\(\newcommand{\vf}[1]{\mathbf{\boldsymbol{\vec{#1}}}}
\renewcommand{\Hat}[1]{\mathbf{\boldsymbol{\hat{#1}}}}
\let\VF=\vf
\let\HAT=\Hat
\newcommand{\Prime}{{}\kern0.5pt'}
\newcommand{\PARTIAL}[2]{{\partial^2#1\over\partial#2^2}}
\newcommand{\Partial}[2]{{\partial#1\over\partial#2}}
\newcommand{\tr}{{\mathrm tr}}
\newcommand{\CC}{{\mathbb C}}
\newcommand{\HH}{{\mathbb H}}
\newcommand{\KK}{{\mathbb K}}
\newcommand{\RR}{{\mathbb R}}
\newcommand{\HR}{{}^*{\mathbb R}}
\renewcommand{\AA}{\vf{A}}
\newcommand{\BB}{\vf{B}}
\newcommand{\CCv}{\vf{C}}
\newcommand{\EE}{\vf{E}}
\newcommand{\FF}{\vf{F}}
\newcommand{\GG}{\vf{G}}
\newcommand{\HHv}{\vf{H}}
\newcommand{\II}{\vf{I}}
\newcommand{\JJ}{\vf{J}}
\newcommand{\KKv}{\vf{Kv}}
\renewcommand{\SS}{\vf{S}}
\renewcommand{\aa}{\VF{a}}
\newcommand{\bb}{\VF{b}}
\newcommand{\ee}{\VF{e}}
\newcommand{\gv}{\VF{g}}
\newcommand{\iv}{\vf{imath}}
\newcommand{\rr}{\VF{r}}
\newcommand{\rrp}{\rr\Prime}
\newcommand{\uu}{\VF{u}}
\newcommand{\vv}{\VF{v}}
\newcommand{\ww}{\VF{w}}
\newcommand{\grad}{\vf{\nabla}}
\newcommand{\zero}{\vf{0}}
\newcommand{\Ihat}{\Hat I}
\newcommand{\Jhat}{\Hat J}
\newcommand{\nn}{\Hat n}
\newcommand{\NN}{\Hat N}
\newcommand{\TT}{\Hat T}
\newcommand{\ihat}{\Hat\imath}
\newcommand{\jhat}{\Hat\jmath}
\newcommand{\khat}{\Hat k}
\newcommand{\nhat}{\Hat n}
\newcommand{\rhat}{\HAT r}
\newcommand{\shat}{\HAT s}
\newcommand{\xhat}{\Hat x}
\newcommand{\yhat}{\Hat y}
\newcommand{\zhat}{\Hat z}
\newcommand{\that}{\Hat\theta}
\newcommand{\phat}{\Hat\phi}
\newcommand{\LL}{\mathcal{L}}
\newcommand{\DD}[1]{D_{\textrm{$#1$}}}
\newcommand{\bra}[1]{\langle#1|}
\newcommand{\ket}[1]{|#1/rangle}
\newcommand{\braket}[2]{\langle#1|#2\rangle}
\newcommand{\LargeMath}[1]{\hbox{\large$#1$}}
\newcommand{\INT}{\LargeMath{\int}}
\newcommand{\OINT}{\LargeMath{\oint}}
\newcommand{\LINT}{\mathop{\INT}\limits_C}
\newcommand{\Int}{\int\limits}
\newcommand{\dint}{\mathchoice{\int\!\!\!\int}{\int\!\!\int}{}{}}
\newcommand{\tint}{\int\!\!\!\int\!\!\!\int}
\newcommand{\DInt}[1]{\int\!\!\!\!\int\limits_{#1~~}}
\newcommand{\TInt}[1]{\int\!\!\!\int\limits_{#1}\!\!\!\int}
\newcommand{\Bint}{\TInt{B}}
\newcommand{\Dint}{\DInt{D}}
\newcommand{\Eint}{\TInt{E}}
\newcommand{\Lint}{\int\limits_C}
\newcommand{\Oint}{\oint\limits_C}
\newcommand{\Rint}{\DInt{R}}
\newcommand{\Sint}{\int\limits_S}
\newcommand{\Item}{\smallskip\item{$\bullet$}}
\newcommand{\LeftB}{\vector(-1,-2){25}}
\newcommand{\RightB}{\vector(1,-2){25}}
\newcommand{\DownB}{\vector(0,-1){60}}
\newcommand{\DLeft}{\vector(-1,-1){60}}
\newcommand{\DRight}{\vector(1,-1){60}}
\newcommand{\Left}{\vector(-1,-1){50}}
\newcommand{\Down}{\vector(0,-1){50}}
\newcommand{\Right}{\vector(1,-1){50}}
\newcommand{\ILeft}{\vector(1,1){50}}
\newcommand{\IRight}{\vector(-1,1){50}}
\newcommand{\Partials}[3]
{\displaystyle{\partial^2#1\over\partial#2\,\partial#3}}
\newcommand{\Jacobian}[4]{\frac{\partial(#1,#2)}{\partial(#3,#4)}}
\newcommand{\JACOBIAN}[6]{\frac{\partial(#1,#2,#3)}{\partial(#4,#5,#6)}}
\newcommand{\LLv}{\vf{L}}
\newcommand{\OOb}{\boldsymbol{O}}
\newcommand{\PPv}{\vf{P}_\text{cm}}
\newcommand{\RRv}{\vf{R}_\text{cm}}
\newcommand{\ff}{\vf{f}}
\newcommand{\pp}{\vf{p}}
\newcommand{\tauv}{\vf{\tau}}
\newcommand{\Lap}{\nabla^2}
\newcommand{\Hop}{H_\text{op}}
\newcommand{\Lop}{L_\text{op}}
\newcommand{\Hhat}{\hat{H}}
\newcommand{\Lhat}{\hat{L}}
\newcommand{\defeq}{\overset{\rm def}{=}}
\newcommand{\absm}{\vert m\vert}
\newcommand{\ii}{\ihat}
\newcommand{\jj}{\jhat}
\newcommand{\kk}{\khat}
\newcommand{\dS}{dS}
\newcommand{\dA}{dA}
\newcommand{\dV}{d\tau}
\renewcommand{\ii}{\xhat}
\renewcommand{\jj}{\yhat}
\renewcommand{\kk}{\zhat}
\newcommand{\lt}{<}
\newcommand{\gt}{>}
\newcommand{\amp}{&}
\definecolor{fillinmathshade}{gray}{0.9}
\newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}}
\)
Section 5.3 Properties of Hermitian Matrices
The eigenvalues and eigenvectors of Hermitian matrices
Section 5.2 have some special properties.
The eigenvalues of a Hermitian matrix must be real.
To see why this relationship holds, start with the eigenvector equation
\begin{equation}
M |v\rangle = \lambda |v\rangle\tag{5.3.1}
\end{equation}
and multiply on the left by \(\langle v|\) (that is, by \(v^\dagger\) ):
\begin{equation}
\langle v | M | v \rangle
= \langle v | \lambda | v \rangle
= \lambda \langle v | v\rangle\text{.}\tag{5.3.2}
\end{equation}
But we can also compute the Hermitian conjugate (that is, the conjugate transpose) of
(5.3.1) , which is
\begin{equation}
\langle v| M^\dagger = \langle v| \lambda^*\text{.}\tag{5.3.3}
\end{equation}
Using the fact that \(M^\dagger=M\text{,}\) and multiplying by \(|v\rangle\) on the right now yields
\begin{equation}
\langle v | M | v \rangle
= \langle v | \lambda^* | v \rangle
= \lambda^* \langle v | v \rangle\text{.}\tag{5.3.4}
\end{equation}
\begin{equation}
\lambda^* = \lambda\tag{5.3.5}
\end{equation}
as claimed.
Eigenvectors corresponding to different eigenvalues must be orthogonal.
The argument establishing this relationship is similar to the one above. Suppose that
\begin{align}
M |v\rangle \amp = \lambda |v\rangle ,\tag{5.3.6}\\
M |w\rangle \amp = \mu |w\rangle\text{.}\tag{5.3.7}
\end{align}
Then
\begin{equation}
\langle v | \lambda | w \rangle
= \langle v | M | w \rangle
= \langle v | \mu | w \rangle\tag{5.3.8}
\end{equation}
or equivalently
\begin{equation}
(\lambda - \mu) \langle v | w \rangle = 0\text{.}\tag{5.3.9}
\end{equation}
Thus, if \(\lambda\ne\mu\text{,}\) \(|v\rangle\) must be orthogonal to \(|w\rangle\text{.}\)
Repeated Eigenvalues.
In the case of repeated eigenvalues, it is possible to make a basis orthogonal using a process called Gram-Schmidt orthogonalization . (This amounts to simply subtracting off the parts of a given basis that are not orthogonal.)
To Remember.
It is always possible to find an orthonormal basis of eigenvectors for any Hermitian matrix.
Sensemaking 5.1 . Properties of Anti-Hermitian Matrices.
Repeat the two proofs above to find similar properties for anti-Hermitian matrices.
Hint 1 . Don’t forget to use \(M^{\dagger}=-M\text{.}\)
Hint 2 . Don’t forget to complex conjugate the eigenvalue, where necessary.
Answer . The eigenvalues are pure imaginary. The eigenvectors corresponding to different eigenvalues are orthogonal.